Classical Hurwitz numbers and related combinatorics

@article{Dubrovin2016ClassicalHN,
  title={Classical Hurwitz numbers and related combinatorics},
  author={Boris Dubrovin and Di Yang and Don Zagier},
  journal={Moscow Mathematical Journal},
  year={2016},
  volume={17},
  pages={601-633}
}
In 1891 Hurwitz [30] studied the number Hg,d of genus g ≥ 0 and degree d ≥ 1 coverings of the Riemann sphere with 2g + 2d− 2 fixed branch points and in particular found a closed formula for Hg,d for any fixed d. These Hurwitz numbers are now very famous and have been studied from many different points of view (matrix models, Gromov–Witten invariants, topological recursion, classical/quantum integrable systems, ...). In this paper we study their combinatorial properties, and compare them with… 

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